To put it in old school terms-
(1)You can either just RP everything, in which case charisma is a useless dump stat and everything is determined by DM fiat as they judge your performance; or
(2) you can use a resolution mechanic for social events, in which case RP is just a mechanic.
There isn't any "right" answer, but these are decidedly different approaches.
I'm not sure that there is a "right" answer (that is, there is probably more than one good way to do things), but I do think that there are wrong answers.
In any event, assuming that both of those are right answers, I think that they are also a false dichotomy. It's not true that either everything is determined by DM fiat or else RP is just a mechanic. There are definitely ways to both engage in RP and also have some system for resolving social tests that doesn't depend only on DM fiat. The real question is just how much prep time do you want to engage in in order to minimize fiat during the run time (fiat during the preparation time, such as what monster a room holds is pretty much impossible to avoid), or which gives you a really great return on investment. My experience on this is for most things preparation time in structuring a social test is better spent elsewhere unless the social test is going to involve some sort of multi-session minigame focused on RP.
But, now moving up to your thief example, the reason you tended to see only Thieves hiding in Shadows is the relatively poor construction of the 1e AD&D rules. There is a tension in the rules. A good rule set tends to have as a meta-rule "Everything that is not forbidden is permitted." However, if you don't outline a good majority of the things that are permitted, there will be a tendency for players to not even try them, simply because they won't be prompted to consider the option. Likewise, if you don't outline a good majority of the things that are permitted, then DMs will tend to find themselves in a bind when propositions don't have a rule that covers them, and the result is likely to be either bad rulesmithing that makes the task too hard or too easy, or simply just saying "No" when they realize too permissive of rulings tend to be vastly more destructive to the game than too restrictive of rulings.
So, in a sense, climbing a wall had always been permitted. But I'm guessing in practice that prior to the introduction of the thief, a given wall was only climbable if the DM called it out as climbable in his own preparation, by for example noting that handholds could be found if the north wall was closely inspected. The thief allowed a player to propose climbing a more or less sheer wall regardless of whether the DM had called out whether it was climbable. Indeed, the introduction of the thief probably started provoking DMs to do the opposite - calling on it in their preparation when a wall was especially not climbable. And this latter habit would tend to make most walls unclimbable except by thieves unless the DM was of a particularly imaginative sort.
There were two other things that created problems. First, the system didn't define what a non-thief could do, which meant it was always up to the DM to decide on some number for the chance for a non-thief. And secondly, and this was a fundamental problem with the thief itself, the thief skills at low level already had such a low chance of success that a good thief player basically never used them anyway, since to face a fortune test was to fail and failure was often lethal. Thus, any number that the DM selected for a non-thief using thief skills would be so low as to be basically saying "No" anyway.
In practice, only M-U's hid in shadows, because they had Invisibility. Thieves had no reliable means of stealth and so rarely utilized it. 'Move Silently' was used, but only because you had to move anyway, so you might as well try to do it moving silently. Ironically though, moving silently had no obvious impact on the Surprise system with its fixed chances of surprise, so unless the DM made some sort of fiat ruling, there was usually little point in doing it.
The point is that we know you don't have to a climb skill that either relies entirely on fiat or else prevents untrained characters from trying to climb, even though we know that poorly thought out implementations might have either consequence.
